Academic Press Handbook Of Mathematical Formula And Integrals 4th Edition Jan 2008 ISBN 0123742889 pdf
Handbook of
Mathematical Formulas and Integrals
FOURTH EDITION
Handbook of
Mathematical Formulas
and Integrals
FOURTH EDITION
Alan Jeffrey Hui-Hui Dai
Professor of Engineering Mathematics Associate Professor of Mathematics University of Newcastle upon Tyne City University of Hong Kong
Newcastle upon Tyne Kowloon, China United Kingdom
AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Academic Press is an imprint of Elsevier Acquisitions Editor: Lauren Schultz Yuhasz Developmental Editor: Mara Vos-Sarmiento Marketing Manager: Leah Ackerson Cover Design: Alisa Andreola Cover Illustration: Dick Hannus Production Project Manager: Sarah M. Hajduk Compositor: diacriTech Cover Printer: Phoenix Color Printer: Sheridan Books Academic Press is an imprint of Elsevier
30 Corporate Drive, Suite 400, Burlington, MA 01803, USA 525 B Street, Suite 1900, San Diego, California 92101-4495, USA
84 Theobald’s Road, London WC1X 8RR, UK This book is printed on acid-free paper.
Copyright c 2008, Elsevier Inc. All rights reserved.
No part of this publication may be reproduced or transmitted in any form or by any
means, electronic or mechanical, including photocopy, recording, or any information
storage and retrieval system, without permission in writing from the publisher. Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone: (+44) 1865 843830, fax: (+44) 1865 853333, E-mail: permissions@elsevier.com. You may also complete your request onlinevia the Elsevier homepage (http://elsevier.com), by selecting “Support & Contact”
then “Copyright and Permission” and then “Obtaining Permissions.” Library of Congress Cataloging-in-Publication Data Application Submitted British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.ISBN: 978-0-12-374288-9 For information on all Academic Press publications visit our Web site at www.books.elsevier.com Printed in the United States of America 08 09 10 9 8 7 6 5 4 3 2 1
Contents
Preface xix
Preface to the Fourth Edition xxi
Notes for Handbook Users xxiii
Index of Special Functions and Notations xliii Quick Reference List of Frequently Used Data
1
0.1. Useful Identities
1
0.1.1. Trigonometric Identities
1
0.1.2. Hyperbolic Identities
2
0.2. Complex Relationships
2
0.3. Constants, Binomial Coefficients and the Pochhammer Symbol
3
0.4. Derivatives of Elementary Functions
3
0.5. Rules of Differentiation and Integration
4
0.6. Standard Integrals
4
0.7. Standard Series
10
0.8. Geometry
12
1 Numerical, Algebraic, and Analytical Results for Series and Calculus
27
1.1. Algebraic Results Involving Real and Complex Numbers
27
1.1.1. Complex Numbers
27
1.1.2. Algebraic Inequalities Involving Real and Complex Numbers
28
1.2. Finite Sums
32
1.2.1. The Binomial Theorem for Positive Integral Exponents
32
1.2.2. Arithmetic, Geometric, and Arithmetic–Geometric Series
36
1.2.3. Sums of Powers of Integers
36
1.2.4. Proof by Mathematical Induction
38
1.3. Bernoulli and Euler Numbers and Polynomials
40
1.3.1. Bernoulli and Euler Numbers
40
1.3.2. Bernoulli and Euler Polynomials
46
1.3.3. The Euler–Maclaurin Summation Formula
48
1.3.4. Accelerating the Convergence of Alternating Series
49
1.4. Determinants
50
1.4.1. Expansion of Second- and Third-Order Determinants
50
1.4.2. Minors, Cofactors, and the Laplace Expansion
51
1.4.3. Basic Properties of Determinants
53 vi Contents
1.4.4. Jacobi’s Theorem
1.11. Power Series
86
1.12.1. Definition and Forms of Remainder Term
86
1.12. Taylor Series
82
1.11.1. Definition
82
79
88
1.10.1. Uniform Convergence
79
1.10. Functional Series
78
1.9.2. Examples of Infinite Products
77
1.9.1. Convergence of Infinite Products
77
1.12.2. Order Notation (Big O and Little o)
1.13. Fourier Series
74
95
1.15.5. Integration of Rational Functions 103
1.15.4. Improper Integrals 101
99
1.15.3. Reduction Formulas
96
1.15.2. Integration
95
1.15.1. Rules for Differentiation
1.15. Basic Results from the Calculus
89
94
1.14.2. Definition and Properties of Asymptotic Series
93
1.14.1. Introduction
93
1.14. Asymptotic Expansions
89
1.13.1. Definitions
1.9. Infinite Products
1.8.3. Examples of Infinite Numerical Series
53
57
1.5.3. Differentiation and Integration of Matrices
62
1.5.2. Quadratic Forms
58
1.5.1. Special Matrices
58
1.5. Matrices
1.4.9. Routh–Hurwitz Theorem
1.5.4. The Matrix Exponential
55
1.4.8. Some Special Determinants
55
1.4.7. Cramer’s Rule
54
1.4.6. Hadamard’s Inequality
54
1.4.5. Hadamard’s Theorem
64
65
72
1.7.1. Rational Functions
1.8.2. Convergence Tests
72
1.8.1. Types of Convergence of Numerical Series
72
1.8. Convergence of Series
69
1.7.2. Method of Undetermined Coefficients
68
68
1.5.5. The Gerschgorin Circle Theorem
1.7. Partial Fraction Decomposition
68
1.6.2. Combinations
67
1.6.1. Permutations
67
1.6. Permutations and Combinations
67
1.15.6. Elementary Applications of Definite Integrals 104 Contents vii
2 Functions and Identities 109
4.2.1. Integrands Involving x
3.1. Derivatives of Algebraic, Logarithmic, and Exponential Functions 149
3.2. Derivatives of Trigonometric Functions 150
3.3. Derivatives of Inverse Trigonometric Functions 150
3.4. Derivatives of Hyperbolic Functions 151
3.5. Derivatives of Inverse Hyperbolic Functions 152
4 Indefinite Integrals of Algebraic Functions 153
4.1. Algebraic and Transcendental Functions 153
4.1.1. Definitions 153
4.2. Indefinite Integrals of Rational Functions 154
n
2.9.3. Trigonometric and Hyperbolic Functions 148
154
4.2.2. Integrands Involving a + bx 154
4.2.3. Integrands Involving Linear Factors 157
4.2.4. Integrands Involving a
2
± b
2
x
2
158
3 Derivatives of Elementary Functions 149
2.9.2. Exponential Functions 147
2.1. Complex Numbers and Trigonometric and Hyperbolic Functions 109
2.5.1. Hyperbolic Functions 132
2.1.1. Basic Results 109
2.2. Logorithms and Exponentials 121
2.2.1. Basic Functional Relationships 121
2.2.2. The Number e 123
2.3. The Exponential Function 123
2.3.1. Series Representations 123
2.4. Trigonometric Identities 124
2.4.1. Trigonometric Functions 124
2.5. Hyperbolic Identities 132
2.6. The Logarithm 137
2.9.1. Logarithmic Functions 147
2.6.1. Series Representations 137
2.7. Inverse Trigonometric and Hyperbolic Functions 139
2.7.1. Domains of Definition and Principal Values 139
2.7.2. Functional Relations 139
2.8. Series Representations of Trigonometric and Hyperbolic Functions 144
2.8.1. Trigonometric Functions 144
2.8.2. Hyperbolic Functions 145
2.8.3. Inverse Trigonometric Functions 146
2.8.4. Inverse Hyperbolic Functions 146
2.9. Useful Limiting Values and Inequalities Involving Elementary Functions 147
2 viii Contents
3
4.2.6. Integrands Involving a + bx 164
4
4.2.7. Integrands Involving a + bx 165
4.3. Nonrational Algebraic Functions 166
√
k
4.3.1. Integrands Containing a + bx and x 166
1 /2
4.3.2. Integrands Containing (a + bx) 168
1 /2
2
4.3.3. Integrands Containing (a + cx ) 170
1 /2
2
4.3.4. Integrands Containing a + bx + cx 172
5 Indefinite Integrals of Exponential Functions 175
5.1. Basic Results 175
ax
5.1.1. Indefinite Integrals Involving e 175
5.1.2. Integrals Involving the Exponential Functions Combined with Rational Functions of x 175
5.1.3. Integrands Involving the Exponential Functions Combined with Trigonometric Functions 177
6 Indefinite Integrals of Logarithmic Functions 181
6.1. Combinations of Logarithms and Polynomials 181
6.1.1. The Logarithm 181
6.1.2. Integrands Involving Combinations of ln(ax) and Powers of x 182
n m
6.1.3. Integrands Involving (a + bx) ln x 183
2
2
6.1.4. Integrands Involving ln(x ) 185 ± a
1 /2 m
2
2
6.1.5. Integrands Involving x ln x + x 186 ± a
7 Indefinite Integrals of Hyperbolic Functions 189
7.1. Basic Results 189
7.1.1. Integrands Involving sinh(a + bx) and cosh(a + bx) 189
7.2. Integrands Involving Powers of sinh(bx) or cosh(bx) 190
7.2.1. Integrands Involving Powers of sinh(bx) 190
7.2.2. Integrands Involving Powers of cosh(bx) 190
m m
7.3. Integrands Involving (a + bx) sinh(cx) or (a + bx) cosh(cx) 191
7.3.1. General Results 191
m n m n
7.4. Integrands Involving x sinh x or x cosh x 193
m n
7.4.1. Integrands Involving x sinh x 193
m n
7.4.2. Integrands Involving x cosh x 193
m n m n
7.5. Integrands Involving x sinh x or x cosh x 193
m n
7.5.1. Integrands Involving x sinh x 193
m n
7.5.2. Integrands Involving x cosh x 194
−m
7.6.195 Integrands Involving (1 ± cosh x)
−1 7.6.1.
195 Integrands Involving (1 ± cosh x)
−2 Contents ix
7.7. Integrands Involving sinh(ax) cosh
n
−m
sin
n
9.2.3. Integrands Involving x
x 210
m
sin
−n
9.2.2. Integrands Involving x
x 209
m
sin
9.2.1. Integrands Involving x
9.2.4. Integrands Involving x
9.2. Integrands Involving Powers of x and Powers of sin x or cos x 209
9.1.1. Simplification by Means of Substitutions 207
9.1. Basic Results 207
9 Indefinite Integrals of Trigonometric Functions 207
arccoth(x/a) 205
−n
8.4.2. Integrands Involving x
arctanh(x/a) 205
−n
8.4.1. Integrands Involving x
arccoth(x/a) 205
−n
x 211
n
−n
sin x/(a + b cos x)
x/(tan x ± 1) 215
n
x or tan
n
9.3.1. Integrands Involving tan
9.3. Integrands Involving tan x and/or cot x 215
214
m
cos x/(a + b sin x)
n
or x
m
n
cos
9.2.7. Integrands Involving x
x 213
−m
cos
n
9.2.6. Integrands Involving x
x 213
m
cos
−n
9.2.5. Integrands Involving x
x 212
m
arctanh(x/a) or x
8.4. Integrands Involving x
−n
p
7.10.1. Integrands Involving (a + bx)
cosh kx 199
m
sinh kx or (a + bx)
m
7.10. Integrands Involving (a + bx)
7.9.2. Integrands Involving coth kx 198
7.9.1. Integrands Involving tanh kx 198
7.9. Integrands Involving tanh kx and coth kx 198
x 197
q
x cosh
7.8.3. Integrands Involving sinh
sinh kx 199
7.8.2. Special Case a = c 197
7.8.1. General Case 196
7.8. Integrands Involving sinh(ax + b) and cosh(cx + d) 196
x 196
n
7.7.2. Integrands Involving cosh(ax) sinh
x 195
n
7.7.1. Integrands Involving sinh(ax) cosh
x 195
−n
x or cosh(ax) sinh
m
7.10.2. Integrands Involving (a + bx)
arccoth(x/a) 204
8.2.2. Integrands Involving x
n
8.3.2. Integrands Involving x
arctanh(x/a) 204
n
8.3.1. Integrands Involving x
arccoth(x/a) 204
n
arctanh(x/a) or x
n
8.3. Integrands Involving x
arccosh(x/a) 203
−n
arcsinh(x/a) 202
m
−n
8.2.1. Integrands Involving x
arccosh(x/a) 202
−n
arcsinh(x/a) or x
−n
8.2. Integrands Involving x
and arcsinh(x/a) or arc(x/c) 201
n
8.1.1. Integrands Involving Products of x
8.1. Basic Results 201
8 Indefinite Integrals Involving Inverse Hyperbolic Functions 201
cosh kx 199
n x Contents
9.4. Integrands Involving sin x and cos x 217
10.1.4. Integrands Involving x
−n
10.1.6. Integrands Involving x
arctan(x/a) 227
n
10.1.5. Integrands Involving x
arccos(x/a) 227
−n
(x/a) 226
10.1.7. Integrands Involving x
m
arccos
n
10.1.3. Integrands Involving x
10.1.2. Integrands Involving x arcsin(x/a) 226
(x/a) 225
m
arcsin
arctan(x/a) 227
n
10.1.1. Integrands Involving x
11.1.2. Special Properties of Ŵ(x) 232
12 Elliptic Integrals and Functions 241
11.1.9. The Incomplete Gamma Function 236
11.1.8. Graph of Ŵ(x) and Tabular Values of Ŵ(x) and ln Ŵ(x) 235
11.1.7. The Beta Function 235
11.1.6. The Psi (Digamma) Function 234
11.1.5. The Gamma Function in the Complex Plane 233
11.1.4. Special Values of Ŵ(x) 233
11.1.3. Asymptotic Representations of Ŵ(x) and n! 233
11.1.1. Definitions and Notation 231
arccot(x/a) 228
231
11.1. The Euler Integral Limit and Infinite Product Representations for the Gamma Function Ŵ(x). The Incomplete Gamma Functions Ŵ (α, x) and γ(α, x)
231
11 The Gamma, Beta, Pi, and Psi Functions, and the Incomplete Gamma Functions
10.1.9. Integrands Involving Products of Rational Functions and arccot(x/a) 229
arccot(x/a) 228
−n
10.1.8. Integrands Involving x
n
225
9.4.1. Integrands Involving sin
x 218
x/ sin
m
x cos
n
x/ cos
m
9.4.4. Integrands Involving sin
−n
x 218
9.4.3. Integrands Involving cos
x 217
−n
9.4.2. Integrands Involving sin
x 217
n
x cos
m
n
9.4.5. Integrands Involving sin
10.1. Integrands Involving Powers of x and Powers of Inverse Trigonometric Functions
n
10 Indefinite Integrals of Inverse Trigonometric Functions 225
x 222
m
cos
n
x or x
m
sin
9.5.2. Integrands Involving x
−m
, sin(cx + d), and/or cos(px + q) 221
n
9.5.1. Integrands Involving Products of (ax + b)
221
9.5. Integrands Involving Sines and Cosines with Linear Arguments and Powers of x
x 220
−n
x cos
12.1. Elliptic Integrals 241 Contents xi
12.1.2. Tabulations and Trigonometric Series Representations of Complete Elliptic Integrals 243
14.1.1. The Fresnel Integrals 261
n
erfc x 259
13.2.9. Value of i
n
erfc x at zero 259
14 Fresnel Integrals, Sine and Cosine Integrals 261
14.1. Definitions, Series Representations, and Values at Infinity 261
14.1.2. Series Representations 261
13.2.7. Integrals of erfc x 258
14.1.3. Limiting Values as x → ∞ 263
14.2. Definitions, Series Representations, and Values at Infinity 263
14.2.1. Sine and Cosine Integrals 263
14.2.2. Series Representations 263
14.2.3. Limiting Values as x → ∞ 264
15 Definite Integrals 265
15.1. Integrands Involving Powers of x 265
15.2. Integrands Involving Trigonometric Functions 267
13.2.8. Integral and Power Series Representation of i
13.2.6. Derivatives of erf x 258
12.1.3. Tabulations and Trigonometric Series for E(ϕ, k) and F (ϕ, k) 245
13 Probability Distributions and Integrals, and the Error Function 253
12.2. Jacobian Elliptic Functions 247
12.2.1. The Functions sn u, cn u, and dn u 247
12.2.2. Basic Results 247
12.3. Derivatives and Integrals 249
12.3.1. Derivatives of sn u, cn u, and dn u 249
12.3.2. Integrals Involving sn u, cn u, and dn u 249
12.4. Inverse Jacobian Elliptic Functions 250
12.4.1. Definitions 250
13.1. Distributions 253
13.2.5. Integrals Expressible in Terms of erf x 258
13.1.1. Definitions 253
13.1.2. Power Series Representations (x ≥ 0) 256
13.1.3. Asymptotic Expansions (x ≫ 0) 256
13.2. The Error Function 257
13.2.1. Definitions 257
13.2.2. Power Series Representation 257
13.2.3. Asymptotic Expansion (x ≫ 0) 257
13.2.4. Connection Between P (x) and erf x 258
15.3. Integrands Involving the Exponential Function 270 xii Contents
15.5. Integrands Involving the Logarithmic Function 273
15.6. Integrands Involving the Exponential Integral Ei(x) 274
16 Different Forms of Fourier Series 275
275
16.1. Fourier Series for f (x) on −π ≤ x ≤ π
16.1.1. The Fourier Series 275 276
16.2. Fourier Series for f (x) on −L ≤ x ≤ L
16.2.1. The Fourier Series 276 276
16.3. Fourier Series for f (x) on a ≤ x ≤ b
16.3.1. The Fourier Series 276 277
16.4. Half-Range Fourier Cosine Series for f (x) on 0 ≤ x ≤ π
16.4.1. The Fourier Series 277 277
16.5. Half-Range Fourier Cosine Series for f (x) on 0 ≤ x ≤ L
16.5.1. The Fourier Series 277 278
16.6. Half-Range Fourier Sine Series for f (x) on 0 ≤ x ≤ π
16.6.1. The Fourier Series 278 278
16.7. Half-Range Fourier Sine Series for f (x) on 0 ≤ x ≤ L
16.7.1. The Fourier Series 278 279
16.8. Complex (Exponential) Fourier Series for f (x) on −π ≤ x ≤ π
16.8.1. The Fourier Series 279 279
16.9. Complex (Exponential) Fourier Series for f (x) on −L ≤ x ≤ L
16.9.1. The Fourier Series 279
16.10. Representative Examples of Fourier Series 280
16.11. Fourier Series and Discontinuous Functions 285
16.11.1. Periodic Extensions and Convergence of Fourier Series 285
16.11.2. Applications to Closed-Form Summations of Numerical Series 285
17 Bessel Functions 289
17.1. Bessel’s Differential Equation 289
17.1.1. Different Forms of Bessel’s Equation 289
17.2. Series Expansions for J ν (x) and Y ν (x) 290
17.2.1. Series Expansions for J n (x) and J ν (x) 290
17.2.2. Series Expansions for Y n (x) and Y ν (x) 291
17.2.3. Expansion of sin(x sin θ) and cos(x sin θ) in Terms of Bessel Functions 292
17.3. Bessel Functions of Fractional Order 292
17.3.1. Bessel Functions J (x) 292
±( n+1 /2 )
17.3.2. Bessel Functions Y (x) 293
±( n+1 /2 )
17.4. Asymptotic Representations for Bessel Functions 294
17.4.1. Asymptotic Representations for Large Arguments 294
17.4.2. Asymptotic Representation for Large Orders 294
17.5. Zeros of Bessel Functions 294 Contents xiii
17.6. Bessel’s Modified Equation 294
17.6.1. Different Forms of Bessel’s Modified Equation 294
17.7. Series Expansions for I ν (x) and K ν (x) 297
17.7.1. Series Expansions for I (x) and I ν (x) 297
n
17.7.2. Series Expansions for K (x) and K (x) 298
n
17.8. Modified Bessel Functions of Fractional Order 298
17.8.1. Modified Bessel Functions I (x) 298
±( n+1 /2 )
17.8.2. Modified Bessel Functions K (x) 299
±( n+1 /2 )
17.9. Asymptotic Representations of Modified Bessel Functions 299
17.9.1. Asymptotic Representations for Large Arguments 299
17.10. Relationships Between Bessel Functions 299
17.10.1. Relationships Involving J ν (x) and Y ν (x) 299
17.10.2. Relationships Involving I (x) and K (x) 301 ν ν
17.11. Integral Representations of J (x), I (x), and K (x) 302
n n n
17.11.1. Integral Representations of J n (x) 302
17.12.1. Integrals of J n (x), I n (x), and K n (x) 302
17.13. Definite Integrals Involving Bessel Functions 303
17.13.1. Definite Integrals Involving J (x) and Elementary Functions 303
n
17.14. Spherical Bessel Functions 304
17.14.1. The Differential Equation 304
17.14.2. The Spherical Bessel Function j (x) and y (x) 305
n n
17.14.3. Recurrence Relations 306
17.14.4. Series Representations 306 306
17.14.5. Limiting Values as x→0
17.14.6. Asymptotic Expansions of j (x) and y (x)
n n
When the Order n Is Large 307
17.15. Fourier-Bessel Expansions 307
18 Orthogonal Polynomials 309
18.1. Introduction 309
18.1.1. Definition of a System of Orthogonal Polynomials 309
18.2. Legendre Polynomials P (x) 310
n
18.2.1. Differential Equation Satisfied by P (x) 310
n
18.2.2. Rodrigues’ Formula for P (x) 310
n
18.2.3. Orthogonality Relation for P n (x) 310
18.2.4. Explicit Expressions for P (x) 310
n
18.2.5. Recurrence Relations Satisfied by P (x) 312
n
18.2.6. Generating Function for P (x) 313
n
18.2.7. Legendre Functions of the Second Kind Q (x) 313
n
18.2.8. Definite Integrals Involving P n (x) 315 xiv Contents
18.2.10. Associated Legendre Functions 316
(x) 330
(x) 331
n
18.5.7. Series Expansions of H
(x) 331
n
18.5.6. Generating Function for H
(x) 330
n
18.5.5. Recurrence Relations Satisfied by H
n
n
18.5.4. Explicit Expressions for H
18.5.3. Orthogonality Relation for H n (x) 330
18.5.2. Rodrigues’ Formula for H n (x) 329
18.5.1. Differential Equation Satisfied by H n (x) 329
(x) 329
n
18.5. Hermite Polynomials H
(x) 327
( α) n
18.4.8. Generalized (Associated) Laguerre Polynomials L
18.5.8. Powers of x in Terms of H
(x) 331
n
( α,β) n (x) 333
18.6.9. Asymptotic Formula for P
( α,β) n (x) 334
18.6.8. The Generating Function for P
( α,β) n (x) 334
18.6.7. Recurrence Relation Satisfied by P
( α,β) n (x) 334
18.6.6. Differentiation Formulas for P
( α,β)
n (x) 33318.6.5. Explicit Expressions for P
18.6.4. A Useful Integral Involving P
18.5.9. Definite Integrals 331
( α,β) n (x) 333
18.6.3. Orthogonality Relation for P
( α,β)
n (x) 33318.6.2. Rodrigues’ Formula for P
( α,β) n (x) 333
18.6.1. Differential Equation Satisfied by P
332
( α,β) n (x)
18.6. Jacobi Polynomials P
18.5.10. Asymptotic Expansion for Large n 332
(x) 327
18.4.7. Integrals Involving L
18.2.11. Spherical Harmonics 318
n
18.3.4. Explicit Expressions for T n (x) and U n (x) 321
(x) 320
n
(x) and U
n
18.3.3. Orthogonality Relations for T
(x) 320
n
(x) and U
18.3.2. Rodrigues’ Formulas for T
18.3.6. Generating Functions for T n (x) and U n (x) 325
(x) 320
n
(x) and U
n
18.3.1. Differential Equation Satisfied by T
(x) 320
n
(x) and U
n
18.3. Chebyshev Polynomials T
18.3.5. Recurrence Relations Satisfied by T n (x) and U n (x) 325
18.4. Laguerre Polynomials L
(x) 327
n
n
18.4.6. Generating Function for L
(x) 327
n
18.4.5. Recurrence Relations Satisfied by L
(x) 326
n
in Terms of L
n
(x) and x
18.4.4. Explicit Expressions for L
n
(x) 326
n
18.4.3. Orthogonality Relation for L
(x) 325
n
18.4.2. Rodrigues’ Formula for L
(x) 325
n
18.4.1. Differential Equation Satisfied by L
(x) 325
( α,β) n (x) for Large n 335 ( α,β) Contents xv
19 Laplace Transformation 337
22.1. Introduction 371
21.1.4. Composite Simpson’s Rule (closed type) 364
21.1.5. Newton–Cotes formulas 365
21.1.6. Gaussian Quadrature (open-type) 366
21.1.7. Romberg Integration (closed-type) 367
22 Solutions of Standard Ordinary Differential Equations
371
22.1.1. Basic Definitions 371
21.1.2. Composite Midpoint Rule (open type) 364
22.1.2. Linear Dependence and Independence 371
22.2. Separation of Variables 373
22.3. Linear First-Order Equations 373
22.4. Bernoulli’s Equation 374
22.5. Exact Equations 375
22.6. Homogeneous Equations 376
22.7. Linear Differential Equations 376
21.1.3. Composite Trapezoidal Rule (closed type) 364
21.1.1. Open- and Closed-Type Formulas 363
19.1. Introduction 337
20.1. Introduction 353
19.1.1. Definition of the Laplace Transform 337
19.1.2. Basic Properties of the Laplace Transform 338
19.1.3. The Dirac Delta Function δ(x) 340
19.1.4. Laplace Transform Pairs 340
19.1.5. Solving Initial Value Problems by the Laplace Transform
340
20 Fourier Transforms 353
20.1.1. Fourier Exponential Transform 353
21.1. Classical Methods 363
20.1.2. Basic Properties of the Fourier Transforms 354
20.1.3. Fourier Transform Pairs 355
20.1.4. Fourier Cosine and Sine Transforms 357
20.1.5. Basic Properties of the Fourier Cosine and Sine Transforms
358
20.1.6. Fourier Cosine and Sine Transform Pairs 359
21 Numerical Integration 363
22.8. Constant Coefficient Linear Differential Equations—Homogeneous Case 377 xvi Contents
22.10. Linear Differential Equations—Inhomogeneous Case and the Green’s Function 382
24.1. Curvilinear Coordinates 433
23.8. Integrals of Vector Functions of a Scalar Variable t 426
23.9. Line Integrals 427
23.10. Vector Integral Theorems 428
23.11. A Vector Rate of Change Theorem 431
23.12. Useful Vector Identities and Results 431
24 Systems of Orthogonal Coordinates 433
24.1.1. Basic Definitions 433
23.6. Derivatives of Vector Functions of a Scalar t 423
24.2. Vector Operators in Orthogonal Coordinates 435
24.3. Systems of Orthogonal Coordinates 436
25 Partial Differential Equations and Special Functions 447
25.1. Fundamental Ideas 447
25.1.1. Classification of Equations 447
25.2. Method of Separation of Variables 451
25.2.1. Application to a Hyperbolic Problem 451
23.7. Derivatives of Vector Functions of Several Scalar Variables 425
23.5. Products of Four Vectors 423
22.11. Linear Inhomogeneous Second-Order Equation 389
22.18. Numerical Methods 404
22.12. Determination of Particular Integrals by the Method of Undetermined Coefficients 390
22.13. The Cauchy–Euler Equation 393
22.14. Legendre’s Equation 394
22.15. Bessel’s Equations 394
22.16. Power Series and Frobenius Methods 396
22.17. The Hypergeometric Equation 403
23 Vector Analysis 415
23.4. Triple Products 422
23.1. Scalars and Vectors 415
23.1.1. Basic Definitions 415
23.1.2. Vector Addition and Subtraction 417
23.1.3. Scaling Vectors 418
23.1.4. Vectors in Component Form 419
23.2. Scalar Products 420
23.3. Vector Products 421
25.3. The Sturm–Liouville Problem and Special Functions 456 Contents xvii
25.5. Conservation Equations (Laws) 457
29 Numerical Approximation 499
525 Index
30.5. Some Useful Conformal Mappings 512 Short Classified Reference List
30.4. Boundary Value Problems 511
30.3. Conformal Transformations and Orthogonal Trajectories 510
30.2. Harmonic Conjugates and the Laplace Equation 510
30.1. Analytic Functions and the Cauchy-Riemann Equations 509
30 Conformal Mapping and Boundary Value Problems 509
29.4. Finite Difference Approximations to Ordinary and Partial Derivatives 505
29.3. Pad´e Approximation 503
29.2. Economization of Series 501
29.1.3. Spline Interpolation 500
29.1.2. Lagrange Polynomial Interpolation 500
29.1.1. Linear Interpolation 499
29.1. Introduction 499
28.1. The z-Transform and Transform Pairs 493
25.6. The Method of Characteristics 458
26.1. The Weak Maximum/Minimum Principle for the Heat Equation 473
25.7. Discontinuous Solutions (Shocks) 462
25.8. Similarity Solutions 465
25.9. Burgers’s Equation, the KdV Equation, and the KdVB Equation 467
25.10. The Poisson Integral Formulas 470
25.11. The Riemann Method 471
26 Qualitative Properties of the Heat and Laplace Equation 473
26.2. The Maximum/Minimum Principle for the Laplace Equation 473
28 The z -Transform 493
26.3. Gauss Mean Value Theorem for Harmonic Functions in the Plane 473
26.4. Gauss Mean Value Theorem for Harmonic Functions in Space 474
27 Solutions of Elliptic, Parabolic, and Hyperbolic Equations 475
27.1. Elliptic Equations (The Laplace Equation) 475
27.2. Parabolic Equations (The Heat or Diffusion Equation) 482
27.3. Hyperbolic Equations (Wave Equation) 488
529
Preface
This book contains a collection of general mathematical results, formulas, and integrals that occur throughout applications of mathematics. Many of the entries are based on the updated fifth edition of Gradshteyn and Ryzhik’s ”Tables of Integrals, Series, and Products,” though during the preparation of the book, results were also taken from various other reference works. The material has been arranged in a straightforward manner, and for the convenience of the user a quick reference list of the simplest and most frequently used results is to be found in Chapter 0 at the front of the book. Tab marks have been added to pages to identify the twelve main subject areas into which the entries have been divided and also to indicate the main interconnections that exist between them. Keys to the tab marks are to be found inside the front and back covers.
The Table of Contents at the front of the book is sufficiently detailed to enable rapid location of the section in which a specific entry is to be found, and this information is supplemented by a detailed index at the end of the book. In the chapters listing integrals, instead of displaying them in their canonical form, as is customary in reference works, in order to make the tables more convenient to use, the integrands are presented in the more general form in which they are likely to arise. It is hoped that this will save the user the necessity of reducing a result to a canonical form before consulting the tables. Wherever it might be helpful, material has been added explaining the idea underlying a section or describing simple techniques that are often useful in the application of its results.
Standard notations have been used for functions, and a list of these together with their names and a reference to the section in which they occur or are defined is to be found at the front of the book. As is customary with tables of indefinite integrals, the additive arbitrary constant of integration has always been omitted. The result of an integration may take more than one form, often depending on the method used for its evaluation, so only the most common forms are listed.
A user requiring more extensive tables, or results involving the less familiar special functions, is referred to the short classified reference list at the end of the book. The list contains works the author found to be most useful and which a user is likely to find readily accessible in a library, but it is in no sense a comprehensive bibliography. Further specialist references are to be found in the bibliographies contained in these reference works.
Every effort has been made to ensure the accuracy of these tables and, whenever possible, results have been checked by means of computer symbolic algebra and integration programs, but the final responsibility for errors must rest with the author.
Preface to the Fourth Edition
The preparation of the fourth edition of this handbook provided the opportunity to enlarge the sections on special functions and orthogonal polynomials, as suggested by many users of the third edition. A number of substantial additions have also been made elsewhere, like the enhancement of the description of spherical harmonics, but a major change is the inclusion of a completely new chapter on conformal mapping. Some minor changes that have been made are correcting of a few typographical errors and rearranging the last four chapters of the third edition into a more convenient form. A significant development that occurred during the later stages of preparation of this fourth edition was that my friend and colleague Dr. Hui-Hui Dai joined me as a co-editor.
Chapter 30 on conformal mapping has been included because of its relevance to the solu- tion of the Laplace equation in the plane. To demonstrate the connection with the Laplace equation, the chapter is preceded by a brief introduction that demonstrates the relevance of conformal mapping to the solution of boundary value problems for real harmonic functions in the plane. Chapter 30 contains an extensive atlas of useful mappings that display, in the usual diagrammatic way, how given analytic functions w = f(z) map regions of interest in the complex z-plane onto corresponding regions in the complex w-plane, and conversely. By form- ing composite mappings, the basic atlas of mappings can be extended to more complicated regions than those that have been listed. The development of a typical composite mapping is illustrated by using mappings from the atlas to construct a mapping with the property that a region of complicated shape in the z-plane is mapped onto the much simpler region compris- ing the upper half of the w-plane. By combining this result with the Poisson integral formula, described in another section of the handbook, a boundary value problem for the original, more complicated region can be solved in terms of a corresponding boundary value problem in the simpler region comprising the upper half of the w-plane.
The chapter on ordinary differential equations has been enhanced by the inclusion of mate- rial describing the construction and use of the Green’s function when solving initial and boundary value problems for linear second order ordinary differential equations. More has been added about the properties of the Laplace transform and the Laplace and Fourier con- volution theorems, and the list of Laplace transform pairs has been enlarged. Furthermore, because of their use with special techniques in numerical analysis when solving differential equations, a new section has been included describing the Jacobi orthogonal polynomials. The section on the Poisson integral formulas has also been enlarged, and its use is illustrated by an example. A brief description of the Riemann method for the solution of hyperbolic equations has been included because of the important theoretical role it plays when examining general properties of wave-type equations, such as their domains of dependence.
For the convenience of users, a new feature of the handbook is a CD-ROM that contains the classified lists of integrals found in the book. These lists can be searched manually, and when results of interest have been located, they can be either printed out or used in papers or xxii Preface
worksheets as required. This electronic material is introduced by a set of notes (also included in the following pages) intended to help users of the handbook by drawing attention to different notations and conventions that are in current use. If these are not properly understood, they can cause confusion when results from some other sources are combined with results from this handbook. Typically, confusion can occur when dealing with Laplace’s equation and other second order linear partial differential equations using spherical polar coordinates because of the occurrence of differing notations for the angles involved and also when working with Fourier transforms for which definitions and normalizations differ. Some explanatory notes and examples have also been provided to interpret the meaning and use of the inversion integrals for Laplace and Fourier transforms.
Alan Jeffrey alan.jeffrey@newcastle.ac.uk Hui-Hui Dai mahhdai@math.cityu.edu.hk Notes for Handbook Users
The material contained in the fourth edition of the Handbook of Mathematical Formulas and Integrals was selected because it covers the main areas of mathematics that find frequent use in applied mathematics, physics, engineering, and other subjects that use mathematics. The material contained in the handbook includes, among other topics, algebra, calculus, indefinite and definite integrals, differential equations, integral transforms, and special functions.
For the convenience of the user, the most frequently consulted chapters of the book are to be found on the accompanying CD that allows individual results of interest to be printed out, included in a work sheet, or in a manuscript.